I've been motivated by this question about starting to study mathematics at an unusually advanced age. It would be nice to know examples of people who successfully switched from a very different field into mathematics.

Which well-known mathematicians, past or present, started out as law/art/humanities/business students, but later turned to mathematics? This excludes mathematicians who switched from the sciences or engineering to mathematics such as Raoul Bott.

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$\begingroup$If the question stays open, it should also be CW. Some kind of motivation for the question other than commendable curiosity would also be good$\endgroup$
– Yemon ChoiSep 28 '11 at 0:17

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$\begingroup$Dear Francois, "At the time of Fermat and Cayley" encompasses more than two centuries, as well as two quite different countries. It's probably possible to make a more refined analysis. (Not that this would necessarily bear on the present question.) Best wishes, Matt $\endgroup$
– EmertonSep 28 '11 at 2:07

$\begingroup$Douglas: Closing a question is never permanent! One of the main reasons for closing a question quickly is to prevent irrelevant answers from appearing before the question is edited to its more-or-less final form.$\endgroup$
– François G. Dorais♦Sep 28 '11 at 3:10

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$\begingroup$The question is better but I still think is in danger of being "ahistorical", in that it presupposes a certain context of education and training which probably didn't apply to Famous People Who Will Be/Have Been Mentioned In The Answers (e.g. Fermat). Francois already made this point well$\endgroup$
– Yemon ChoiSep 28 '11 at 19:41

21 Answers
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Persi Diaconis left home at 14 to work with Dai Vernon as a magician. Trying to protect himself from being cheated in dishonest casinos, he was led to Feller's textbook on probability theory, which he couldn't understand. He started studying calculus at the City College of New York at the age of 24. Some more details may be found in this article.

I remember reading an interview with Vladimir Arnold where he tells the following anecdote about Hassler Whitney. I don't have the interview in front of me, so some of the details may be not quite correct, but if memory serves, the story goes on like this.

Whitney used to study music in America and at some point decided to spend a year in Germany. He arrived in G\"ottingen where it turned out that he had to take a course outside his main subject of study, which was music. He asked which one of the courses was the most difficult one. It turned out the most difficult subject was quantum mechanics. Whitney enrolled on that course. After the first lecture he came to see the professor and said

-- I was one of the best students in Yale in my year, Herr Professor; how come I didn't understand a single word of the lecture?"

--Well, you see, there are some prerequisites for this course. You have to know calculus and linear algebra and ....

-- Are there any books where I can read all this up?

It took Whitney a couple of weeks to work through the books the professor told him to read. In a month Whitney was able to follow the course, and he decided to switch to mathematics at the end of the semester.

Arnold tells this story to illustrate the dangers of early specialization.

$\begingroup$I heard about Whitney's starting out in music from late Stanislaw Lojasiewicz (who also once contemplated a career in music), but could not find any confirmation in writing. $\endgroup$
– Margaret FriedlandSep 28 '11 at 21:18

$\begingroup$Margaret -- again if memory serves, Arnold heard the story from Whitney himself; he regretted he hadn't asked who the quantum mechanics lecturer was.$\endgroup$
– algoriSep 29 '11 at 21:58

$\begingroup$Apparently, $\L$ojasiewicz also heard it form Whitney- but I have only a recollection of him relating the story once at a lunch table. It is good that you were able to point to a more reliable reference; the story deserves to be known. I wish I had asked $\L$ojasiewicz about his own background in music.$\endgroup$
– Margaret FriedlandSep 30 '11 at 13:46

Edward Witten.(Fields Medalist) Witten attended the Park School of Baltimore (class of '68), and went on to receive his Bachelor of Arts with a major in history and minor in linguistics from Brandeis University in 1971. He planned to become a political journalist, and published articles in The New Republic and The Nation. In 1968 Witten published an article in The Nation arguing that the New Left had no strategy. He worked briefly for George McGovern, a Democratic presidential nominee in 1972. McGovern lost the election in a landslide to Richard Nixon. look at http://www.youtube.com/watch?v=f1BcyxQCnoE&feature=related

$\begingroup$He had notable achievements in linguistics, too: he translated Rigveda (from Sanskrit to German) and observed some pattern in phonology that became known as Grassman's law.$\endgroup$
– Margaret FriedlandSep 28 '11 at 18:59

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$\begingroup$Grassmann also compiled a dictionary, which people use up to this day to read Rigveda (or at least were using 15 or so years ago).$\endgroup$
– algoriSep 28 '11 at 22:21

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$\begingroup$However, Grassman did his work on Indo-European studies and the Rigveda only after he got disappointed by the fact that his mathematics hadn’t been as accepted by the community as he hoped.$\endgroup$
– k.stmMar 7 '18 at 18:46

"After returning to the United States, Lang went to Princeton University with the intention of studying for a doctorate in philosophy. After a year in the philosophy department, he changed to mathematics and Emil Artin became his thesis advisor."

$\begingroup$Halmos also reports starting out in philosophy, but he doesn't say he "finished it". The way he tells it (in his automathography), it was more like it finished him, after a disappointing qualifying exam, and he switched to mathematics. (Much to the benefit of the latter.) (Oh, and oops, I just noticed the answer by Michael Hardy and Did's comment which also gives this information.)$\endgroup$
– Todd Trimble♦Apr 12 '15 at 6:09

Karl Marx. So you didn't know he was a mathematician? A book of his collected mathematical papers is in our math library, which is more than most mathematicians can claim. (They are mostly attempts to understand the definition of a derivative if I recall correctly.) They were quite popular during the cultural revolution, Chinese mathematicians presumably figuring that the study of dialectical calculus was better then a one-way trip to one of Mao's holiday resorts.

$\begingroup$From the current draft of the original question: "Which well-known mathematicians, past or present, started out as law/art/humanities/business students, but later turned to mathematics?"$\endgroup$
– Yemon ChoiSep 28 '11 at 20:45

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$\begingroup$Let us not forget the famous mathematician Napoleon, having a theorem named after him.$\endgroup$
– user9072Sep 28 '11 at 21:04

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$\begingroup$Another example in reverse: Edmund Husserl (he studied with Weierstrass and wrote about foundations of mathematics and logic).$\endgroup$
– Margaret FriedlandSep 28 '11 at 21:15

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$\begingroup$Engels writing to Marx: "Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and was pleased to find that I did not need them. I complement you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes with the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls. . . You need not fear that any mathematician has preceded you here."$\endgroup$
– Steven LandsburgSep 28 '11 at 22:13

$\begingroup$As re Eugenia Cheng above: does this count as someone with a background in "non-maths" turning to maths?$\endgroup$
– Yemon ChoiSep 29 '11 at 21:14

$\begingroup$@Yemon: I think so, but I'm no expert on music so I could be wrong. In the autobiography on his web site he discusses his teenage years, and it seems to me that he was considered a prodigy and accomplished at this time (e.g. had played as soloist with the Royal Opera Orchestra of Sweden and many other concerts, and studied with masters and so on). He also mentions that "I did little or no systematic study of mathematics in these years." He also suggests there that he's never really given piano away, but there is a clear point where mathematics (later) enters the picture.$\endgroup$
– Philip BrookerSep 30 '11 at 4:32

$\begingroup$In any case he started to study math at the university right after high-school at age 18 (like 'everybody else').$\endgroup$
– user9072Sep 30 '11 at 9:36

I love this one: Harald Bohr was such a good soccer player that he was member of the Danish national team at the 1908 Olympiads. Two years later he got his PhD (apparently there was a large crowd at the event, a quite unusual occurrence for the math department) and went on to become a famous mathematician.

$\begingroup$Where is there any switch in fields. He was studying maths since 1904, starting aged 17/18. In addition, he had a hobby at which he was really good.$\endgroup$
– user9072Sep 28 '11 at 21:09

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$\begingroup$I agree this does not qualify as an answer if not marginally. On the other hand, I do not understand your persistence in proving that. Several other answers are even worse! (what about Fermat or Leibnitz? shall we talk about Plato?) The main reason I posted this is that I love this fact snippet, and in particular the scene of a huge hooligan crowd gathering at the discussion of a PhD thesis in math :)$\endgroup$
– Piero D'AnconaSep 29 '11 at 9:00

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$\begingroup$Yes, it is a fun fact; and in some sense more interesting than Leibniz, Fermat, Euler,... in my opinion also more than the music-semi-examples, which are not all that surprising or unique. And both Yemon and I 'complained' in general about the too old examples (in my present this is only implict but I head a more explict one that I deleted not to clutter the general comment thread to much). So, I/we are not singularly 'complaining' about your answer. Perhaps we can agree that yours is off-topic in an fun/interesting/original way.$\endgroup$
– user9072Sep 29 '11 at 16:24

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$\begingroup$Just to echo quid: for all the good it does, which is little, I have downvoted Richard Borcherds's egregiously off-topic answer. But hey, Fields Medallist! Upvote! Fun story! Upvote! [sigh]$\endgroup$
– Yemon ChoiSep 29 '11 at 20:29

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$\begingroup$Ok. Pointless discussion I guess, the original question was not too serious to begin with$\endgroup$
– Piero D'AnconaSep 29 '11 at 22:08

$\begingroup$True (and I'd say that "composer" implies "musician"), but $-$ though I have no direct recollection $-$ I have it on good authority that this is not an example of "starting out" in arts and then "turning to science", because here fascination with numbers came first, albeit by only a few months.$\endgroup$
– Noam D. ElkiesSep 28 '11 at 0:53

$\begingroup$Sorry, when I answered, I only paid attention to the title of this thread, and you certainly count as a mathematician with "background in" the arts.$\endgroup$
– David CorwinSep 5 '12 at 18:15

$\begingroup$@DavidCorwin: and a member of MO!$\endgroup$
– José Hdz. Stgo.Feb 18 '14 at 2:15

$\begingroup$And a world-class composer of chess problems (also, at least at one point, a chess Master), and from the lore that I've heard, quite a respectable weight lifter as well! Life isn't fair...$\endgroup$
– Todd Trimble♦Apr 12 '15 at 6:24

Christiaan Huygens (1629–1695). A musical prodigy who by age 10 read fluently in four clefs and played the organ, two years before his first study of mathematical sciences (though admittedly 12 is not an "unusually advanced age" for that either...). This according to a display case at Leiden University which I saw at the ANTS-IV conference in 2000, and which reproduced some of his harmony exercises!

Huygens kept up his interest in music, later in life publishing a treatise on a tuning of 31 equal notes to the octave, an idea that apparently still has some currency in the Dutch music scene. According to Huygens' Wikipedia entry, the 20th volume of his 22-volume Collected works is titled Musique et mathématique. Musique. Mathématiques de 1666 à 1695.

Daniel Bernoulli. I am not sure if he counts, because he knew he wanted to study mathematics. Yet his formal education was in business and medicine. According to Wikipedia,

"Around schooling age, his father, Johann Bernoulli, encouraged him to study business, there being poor rewards awaiting a mathematician. However, Daniel refused, because he wanted to study mathematics. He later gave in to his father's wish and studied business. His father then asked him to study in medicine, and Daniel agreed under the condition that his father would teach him mathematics privately, which they continued for some time."

Henri Poincaré was a mining engineer. His first job was at the Corps des Mines as an inspector of mines. He participated in the rescue of miners trapped after an explosion, himself descending the shaft into the mine to investigate the cause of the explosion! Check this link for details.

$\begingroup$I've never thought of mining engineering being related to maths. Possibly the world's most famous mining engineer, Major-General Sir Richard Hannay, KCB, OBE, DSO, Legion of Honour, never struck me as the maths-ey type...$\endgroup$
– ADLSep 30 '11 at 10:32

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$\begingroup$@Alan, are you informed about the French higher eductations system (at Poinarés time)? It is not overly surpising that he was in the Corps des Mines. And most high-school math teachers are not research math types either. Shall we include all French math that attended the/a ENS because clearly they originally all wanted to become high-school teachers. And those at the Ecole polytchnique are actually soldiers. $\endgroup$
– user9072Sep 30 '11 at 11:18

$\begingroup$"This excludes mathematicians who switched from the sciences or engineering to mathematics such as Raoul Bott." If not for this provision, one could name even more people, e.g. Solomon Lefschetz.$\endgroup$
– Margaret FriedlandSep 28 '11 at 20:14

$\begingroup$Could you give a link to the interview- an interview with K. Kato sounds quite interesting$\endgroup$
– sisnOct 12 '11 at 7:45